Mini Course on Wave Turbulence | Recent progress on mathematical theory of wave turbulence

Abstract

The theory of wave turbulence is the statistical theory of interacting waves, or the wave analog of Boltzmann's kinetic theory. It started in the works of Peierls in the 1920s, and has had substantial developments in the 20th century, with significant applications in science. The mathematical study of this subject started with the early works of Spohn and Erdös-Yau et al. on linear models, and the nonlinear problem has recently attracted a lot of attention from the PDE community.In a sequence of papers published in the last few years, joint with Zaher Hani (University of Michigan), we have completely settled this full nonlinear problem, and established the rigorous mathematical foundation of the wave turbulence theory. In this mini-course, which contains four 2-hour lectures, I will briefly explain the scope of this subject, historical timeline, the recent works, and ideas involved in the proof. The first two lectures will focus on more general and intuitive descriptions of the theory, and the last two lectures will contain an overview of the important components of the proof.

About the Speaker

Yu Deng 邓煜

邓煜（Yu Deng），北京大学数学科学学院2007级本科校友，2008年秋季转入麻省理工学院。2010年获普特南（Putnam）大学生竞赛最高奖——Putnam Fellow.2015年于普林斯顿大学获得博士学位，随后到纽约大学库朗研究所（Courant Institute）做博士后。2018年加入南加州大学，现任助理教授。

他和他的合作者取得的重要研究成果包括：对二维高次Schrödinger方程证明了Gibbs测度的不变性；对Schrödinger方程对应的弱湍流问题，在最佳的时间尺度上验证了极限方程的正确性；证明了三维水波方程在重力-表面张力作用下的小初值整体存在性；证明了二维Couette flow在超临界空间的不稳定性。